Optical properties of gold and silver spherical plasmonic ...

Institute of Physics Physics, Polish Academy of Sciences Sciences, Warsaw

Optical properties of gold and silver spherical plasmonic nanoantennas: size dependent multipolar resonance frequencies and plasmon damping rates Krystyna Kolwas Mykola Shopa Anastasiya Derkachova Gennadiy Derkachov 1 http://www.ifpan.edu.pl/ON-2/on22/staff/kolwak.html

Institute of Physics Polish Academy of Sciences

„Small is different” different” physical and chemical properties of some nanometer size structures

U. Landmann Example:

gold, the element known as the most noble from all the metals. When reduced in size to the nanometer range, gold looses its nobleness (and colour)

striking reactive features spectacular optical effects unusual electric conductivity unusual surface friction properties mechanical features (e.g. ductility) …

- the least reactive metal

2


Properties of metal nanoparticles depend on:

shape size

This work: • optical properties of spherical particles • the quantitative description of their plasmon-dependent size characteristics. z

q out Rin

Incoming light f

x

striking reactive features spectacular optical effects unusual electric conductivity unusual surface friction properties mechanical features (e.g. ductility) …

y


Size and shape dependence: z q out Rin

Incoming light f

y

x

Scattering of white light by 30nm silver nanoparticles and clusters of nanoparticles of various sizes and shapes (dark-field microscope image )

Image: Neil Anderson

10µm

M. Beversluis , Ph.D.Thesis, University of Rochester, 2005

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Size and shape dependence: z q out Rin

Incoming light f

y

x whitee light illumination: whit

Despite extremely low concentration of gold particles (< 10−2 weight %), suspensions show bright colours, which depend on the size of particles.

back :

φ = 150, 100, 80, 60, 40, 200 nm

front :

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Suspensions of nanospheres


Mie scattering theory G. Mie, Ann. Phys. 25, 377 (1908)

- a plane electromagnetic wave illuminating a homogeneous, perfectly spherical particle embedded in an infinite homogenous host medium - the solutions of Maxwell equations supplemented by appropriate boundary conditions whitee light illumination: whit

Mie theory explains many observed effects, e.g. diversity of colours of gold particles of diverse sizes.

Image: Neil Anderson

10µm

M. Beversluis , Ph.D.Thesis, University of Rochester, 2005

φ = 150, 100, 80, 60, 40, 200 nm

6 C. Sönnichsen, Dissertation der Fakultät für Physik der LudwigMaximilians-Universität München, 2004


Size dependence

WHY these effects effects are so wavelength selective? What is the origin of spectacular colour effects observed?

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Plasmon excitations

z q

The unique optical properties of metal nanostructures are known to be due to surface plasmons. plasmons

in

out

R f

y

x

Surface plasmons:: collective oscillations of free-electrons at the metal-dielectric interface and electromagnetic fields associated with these metaloscillations (TM polarization).

However: Mie theory does not deal with the notion of plasmon resonances. In spite: maxima in the absorption or scattering resulting from Mie theory are interpreted as due to plasmon resonance positions.

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Eigenfrequencies of a metal sphere Our interest: rigorous description of optical properties of a plasmonic spherical particle as a function of size:

ω = ω l ( R ) , l = 1,2,3,... Possibility of resonant excitation of the SP oscillations and damping of these oscillations we describe as the inherent property of a conducting sphere,. The eigenvalue problem is formulated in absence of external, incoming fields and is abstracted from the quantity, which can be observed (analogy to SP at a flat metal–dielectric interface)

z q out Rin

Incoming light f

y

x

central problem: dispersion relation for the wave at the spherical metal–dielectric interface

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K. Kolwas, S. Demianiuk, M. Kolwas, J. Phys. B, 29, (1996) 4761 K. Kolwas, A. Derkachova, S. Demianiuk, Comp.. Mat. Sc., 35, (2006) 337 K. Kolwas, A. Derkachova, M. Shopa J.Q.S.R.T. 110 1490–1501(2009)


Surface plasmons (re-consideration, rigorous size dependent modelling)

Helmholtz equations + continuity relations Flat boundary: Dispersion relation:

Spherical boundary: Equation defining dispersion relation (TM mode):

z - Bessel function of the 1-st and2-nd kind

q

- Hankel function

in

out

R f

y

x R. Fuchs , P. Halevi, Halevi, Spatial Dispertion in Solids and Plasmas, ed. P. Halevi, Halevi, 1992 K. Kolwas, S. Demianiuk, M. Kolwas, J. Phys. B, 29, (1996) 4761 K. Kolwas, A. Derkachova, S. Demianiuk, Comp.. Mat. Sc., 35, (2006) 337


The explicit size dependence of plasmon resonance frequencies - would be very useful in the numerous plasmon applications

that involve e.g. plasmon near field enhancement. enhancement SERS spectroscopic techniques

Guiding the light along the array of nanoparticles

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Institute of Physics, Polish Academy of Sciences, Warsaw

Eigenfrequencies of a metal sphere (rigorous size dependent modelling)

ω p2 ε D (ω ) = ε ∞ − ω (ω + iγ )

relaxation rate of free electron movement

contribution of the bound electrons to the polarizability

Self--consistent Self (divergent free free))

Maxwell eq. (no external fields)

+

Continuity relations for tangent component of E i H at frequency ω

Effective parameters for gold: ε∞ = 9,84eV ωp = 9,09eV γ = 0,072eV

Conditions for existence of solutions (TM in character) define: Ωl (R) = ω’l (R) + iω ω”l (R)

Frequencies of plasmon oscillations oscillations

Damping rates (radiation and heat) 12


The explicit plasmon size characteristics Ωl (R) = ω’l (R) + iω ω”l (R) ω’l (R) – resonace frequencies of plasmon oscillations of multipolar modes l =1,2,3,…,

0≠

|ω”l (R)| - plasmon damping rates (radiation and heat)

When plasmon is excited, a sphere acts as

a reciving or radiating antenna, depending on the relative contribution of • radiative damping rate γlrad(R) and • nonradiative damping rate γ/2 to the total plasmon damping rate:

|ω”l (R)|= γlrad(R) + γ/2 13


Plasmon size characteristics for noble metal nanospheres Plasmon resonance frequencies

Plasmon damping rates (dominated by radiative damping)

Resulting plasmon size characteristics: ready-to-use data allowing to optimize surface plasmon features in practical 14 applications by choosing: • appropriate particle size for choosen application Derkachova, accepted in Opto Opto--Electr. Electr. Rev. (2010 2010)) • nanosphere material and its environment. K. Kolwas, A. Derkachova,


Plasmon size characteristics for noble metal nanospheres Plasmon resonance frequencies

Plasmon damping rates (dominated by radiative damping)

Plasmons contribute to the particle near field distribution and to the particle 15 images formatted in the particle surface proximity and cause: • enhancement of the near field intensity, • but also important modification of the spatial field and intensity distribution.


Near field images of a plasmonic nanosphere (Mie theory) Our aim is to reconstruct :

1.

2.

The image of the particle in the XY detection plane as a function of scanning field frequency (close to and equal to the plasmon dipole and quadrupole resonance frequencies) The particle image and distribution of Er and its modification with changing distance d to the detection plane at the plasmon dipole and quadrupole resonance frequencies: ω =ω’l=1,2 (R)

16 W. Bazhan, K. Kolwas, Comp. Phys. Comm., 165 191 (2005)


Near field images of a plasmonic nanosphere (Mie theory) A plasmonic scattering object: Au sphere of radius 95nm: • •

Significant plasmon radiative rates Dipole and quadrupole plasmon resonance frequencies in the visible range:

ωl=2(95nm)=2,57eV

ωl=1(95nm)=1,96eV

17 K. Kolwas, A. Derkachova, Derkachova, accepted in Opto Opto--Electr. Electr. Rev. (2010 2010))


Near field images of a plasmonic nanosphere (Mie theory) A plasmonic scattering object: Au sphere of radius 95nm: • •

Significant plasmon radiative rates Dipole and quadrupole plasmon resonance frequencies in the visible range

Orthogonal scattering geometry: polarization of the incident and scattered field: • perpendicular • parallel to the observation plane ”Clinical” orthogonal scattering geometries that enable to expose: • a single dipole (l = 1) plasmon contribution, • a single quadrupole (l = 2) plasmon contribution and to separate these contributions spatially from l < 2 contributions by observing I⊥(ω,R) and I||(ω,R) scattered intensities (Mie theory)

18 K. Kolwas, A. Derkachova, J.Q.S.R.T. 110 1490–1501(2009)


Near field images of a plasmonic nanosphere (Mie theory) K. Kolwas, A. Derkachova, M. Shopa J.Q.S.R.T. 110 1490–1501(2009 S. Demianiuk, K. Kolwas.. J. Phys. B, 34 1651 (2001)

Alternatively:

19 K. Kolwas, A. Derkachova, J.Q.S.R.T. 110 1490–1501(2009)


Near field imaging I(x,y)~Sn(x,y) as a function of a distance d between the particle and the scanning plane The Poynting vector S(θ,ϕ, r) for scattered fields: ~

The numerical image of the particle in the XY scanning plane: the distribution of Sn(θ,ϕ, r) over this plane:

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Near field imaging The role of the distribution of the Er component with increasing distance d between the particle and the scanning plane 1 , r2 1 Eθ ,ϕ , H θ ,ϕ ~ r Er , H r ~

The numerical image of the particle in the XY scattering plane: the distribution of Sn(θ,ϕ, r) over this plane:

Er and Hr play a remarkable role in building the near-field image of a plasmonic particle:

S n ~ ( Eθ H ϕ − Eϕ H θ ) cos ϕ1 + ( − E r H ϕ + Eϕ H r ) sin θ cos ϕ + ( E r H θ − Eθ H r ) sin ϕ21


Near field distribution Er(x,y) as a function of a distance d between the particle and the scanning plane Er(x,y)

at the dipole plasmon resonance frequency ωl=1(95nm) =1,96eV (Vv (⊥) observation geometry)

M. Shopa...

http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/er_vv_distance_func.avi

ωl=1(95nm)=1,96eV Er distribution is directly connected to the pattern of the plasma waves induced at the particle surface. Er(x,y)

at the quadrupole plasmon resonance frequency ωl=2(R=95nm) =2,57eV (Hh (||) observation geometry) ωl=2(95nm)=2,57eV

http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/er_hh_distance_func.avi

⊗

d : (R=75) ÷ 1000nm

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Near field imaging I(x,y)~Sn(x,y) as a function of a distance d between the particle and the scanning plane I(x,y)

at the dipole plasmon resonance frequency (Vv (⊥) observation geometry)

M. Shopa...

http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/ivv_distance_func.avi

ωl=1(95nm)=1,96eV

I(x,y)

at the quadrupole plasmon resonance frequency (Hh (||) observation geometry) ωl=2(95nm)=2,57eV

http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/ihh_distance_func.avi

⊗

d : (R=75) ÷ 1000nm

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Near field imaging

I(x,y)~Sn(x,y) as a function of the incoming light wave frequency ω I(x,y)

M. Shopa...

http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/ivv_1,2_3,2ev.avi

Vv (⊥) observation geometry ωl=1(95nm)=1,96eV

ω scanned: 1,2eV ÷ 3,2eV I(x,y) http://www.ifpan.edu.pl/ON2/on22/films/near_field_immaging_plane/ihh_1,2_3,2ev.avi

Hh (||) observation geometry ωl=2(95nm)=2,57eV

⊗ 24

at once

d = 95nm=R


Summary 1. Eigenvalue problem for plasmonic particle: find the explicit multipolar plasmon size characteristics • within rigorous size dependent modelling • for realistic, frequency dependent dielectric function of a metal εin(ω)

2. Imaging of a plasmonic particle: • classify (and understand) diverse near-field images of a plasmonic spherical particle • at frequencies close to and equal to the frequency of dipole and quadrupole plasmon and • understand, what localization of the surface plasmon means.

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Thank you for your attention Krystyna Kolwas26